Language Reference

PRODUCT Function

multiplies matrices of polynomials

PRODUCT( a, b\lt, dim>)

The inputs to the PRODUCT function are as follows:


a
is an m x (ns) numeric matrix. The first m x n submatrix contains the constant terms of the polynomials, the second m x n submatrix contains the first-order terms, and so on.

b
is an n x (pt) matrix. The first n x p submatrix contains the constant terms of the polynomials, the second n x p submatrix contains the first-order terms, and so on.

dim
is a 1 x 1 matrix, with value p\gt. The value of this matrix is used to set the dimension p of the matrix b. If omitted, the value of p is set to 1.
The PRODUCT function multiplies matrices of polynomials. The value returned is the m x (p(s+t-1)) matrix of the polynomial products. The first m x p submatrix contains the constant terms, the second m x p submatrix contains the first-order terms, and so on.

Note: The PRODUCT function can be used to multiply the matrix operators employed in a multivariate time series model of the form
\phi_1(b) \phi_2(b) y_t = \theta_1(b) \theta_2(b) \epsilon_t
where \phi_1(b), \phi_2(b), \theta_1(b), and \theta_2(b) are matrix polynomial operators whose first matrix coefficients are identity matrices. Often \phi_2(b) and \theta_2(b) represent seasonal components that are isolated in the modeling process but multiplied with the other operators when forming predictors or estimating parameters. The RATIO function is often employed in a time series context as well.

For example, the following statement produces the matrix r, as shown:

  
    r=product({1 2 3 4, 
               5 6 7 8}, 
              {1 2 3, 
               4 5 6}, 1);
 

  
                 R             2 rows      4 cols    (numeric) 
  
                            9        31        41        33 
                           29        79       105        69
 

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